A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

On a multilinear singular integral operator

In this paper, we prove that the maximal operatorsatisfiesis homogeneous of degree 0, has vanishing moment up to order M and satisfies Lq-Dini condition for some     

Randomly weighted sums of dependent subexponential random variables*

We consider the randomly weighted sums $ \sum\nolimits_{k = 1}^n {{\theta_k}{X_k},n \geqslant 1} $ , where $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ are n real-valued random variables with subexponential distributions, and $ \left\{ {{\theta_k},1 \leqslant k \leqslant n} \right\} $ are other n random variables independent of $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ and satisfying $ a \leqslant \theta \leqslant b $ for some $ 0 < a \leqslant b < \infty $ and all $ 1 \leqslant k \leqslant n $ . For $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ satisfying some dependent structures, we prove that $$ {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant m \leqslant n} \sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant k \leqslant n} {\theta_k}{X_k} > x} \right)\sim \sum\limits_{k = 1}^m {{\text{P}}\left( {{\theta_k}{X_k} > x} \right)} $$ as x??????.     

A discreteness criterion for the spectrum of a quasielliptic operator

For the spectrum of the operator $$u = \sum\nolimits_{j = 1}^n {( - 1)^{m_j } D_j^{2m_j } u + q(x)u,} $$ to be discrete, where the m_j are arbitrary positive integers such that \(\sum\nolimits_{j = 1}^n {\tfrac{1}{{2m_j }}< 1} \) , and q(x) ≥ 1, it is necessary and sufficient that \(\int\limits_K {q (x) dx \to \infty } \) , when the cube K tends to infinity while preserving its dimensions.     

Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions

One of the principal topics of this paper concerns the realization of self-adjoint operators L _Θ,Ω in L ²(Ω; d ⁿ x) ^m , m, n ∈ ?, associated with divergence form elliptic partial differential expressions L with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains Ω ? ? ⁿ . In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions L which act as $$Lu = - \left( {\sum\limits_{j,k = 1}^n {\partial _j } \left( {\sum\limits_{\beta = 1}^m {a_{j,k}^{\alpha ,\beta } \partial _k u_\beta } } \right)} \right)_{1 \leqslant \alpha \leqslant m} , u = \left( {u_1 , \ldots ,u_m } \right).$$ The (nonlocal) Robin-type boundary conditions are then of the form $$v \cdot ADu + \Theta [u|_{\partial \Omega } ] = 0{\text{ on }}\partial \Omega ,$$ where Θ represents an appropriate operator acting on Sobolev spaces associated with the boundary ?Ω of Ω, ν denotes the outward pointing normal unit vector on ?Ω, and $Du: = \left( {\partial _j u_\alpha } \right)_{_{1 \leqslant j \leqslant n}^{1 \leqslant \alpha \leqslant m} } .$ Assuming Θ ≥ 0 in the scalar case m = 1, we prove Gaussian heat kernel bounds for L _Θ,Ω, by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on ?Ω. We also discuss additional zero-order potential coefficients V and hence operators corresponding to the form sum L _Θ,Ω + V.     

Best approximation and de la Vallée-Poussin sums

For the class C_ε={f∈C_2π: E_n, n≤Z₊} where \(\left\{ {\varepsilon _n } \right\}_{n \in Z_ + } \) is a sequence of numbers tending monotonically to zero, we establish the following precise (in the sense of order) bounds for the error of approximation by de la Vallée-Poussin sums: (1) $$c_1 \sum\nolimits_{j = n}^{2\left( {n + l} \right)} {\frac{{\varepsilon _j }}{{l + j - n + 1}}} \leqslant \mathop {\sup }\limits_{f \in C_\varepsilon } \left\| {f - V_{n, l} \left( f \right)} \right\|_C \leqslant c_2 \sum\nolimits_{j = n}^{2\left( {n + l} \right)} {\frac{{\varepsilon _j }}{{l + j - n + 1}}} \left( {n \in N} \right)$$ , where c₁ and c₂ are constants which do not depend on n orl. This solves the problem posed by S. B. Stechkin at the Conference on Approximation Theory (Bonn, 1976) and permits a unified treatment of many earlier results obtained only for special classes C_ε of (differentiable) functions. The result (1) substantially refines the estimate (see [1]) (2) $$\left\| {V_{n, l} \left( f \right) - f} \right\|_C = O\left( {\log {n \mathord{\left/ {\vphantom {n {\left( {l + 1} \right) + 1}}} \right. \kern-\nulldelimiterspace} {\left( {l + 1} \right) + 1}}} \right) E_n \left[ f \right] \left( {n \to \infty } \right)$$ and includes as particular cases the estimates of approximations by Fejér sums (see [2]) and by Fourier sums (see [3]).     

Some boundedness results for systems of two rational difference equations

We study k ^th order systems of two rational difference equations

Unboundedness results for systems

We study k ^th order systems of two rational difference equations

The algebraic independence of certain transcendental continued fractions

In the present note the algebraic independence of certain continued fractions is proved. Especially, we prove that the Böhmer-Mahler's series \(\sum\limits_{K = 1}^\infty {\left[ {\omega _v k} \right]} {\text{ }}g_\mu ^{ - k} \left( {1 \leqslant \mu \leqslant s,1 \leqslant v \leqslant t} \right)\) are algebraically independent, where \(\mathop \omega \nolimits_1 {\text{ , }}...{\text{ , }}\mathop \omega \nolimits_{\text{t}} \) , ..., \(\mathop g\nolimits_1 {\text{ , }}...{\text{ , }}\mathop g\nolimits_s \) are some irrational numbers andg ₁, ...,g _s are distinct positive integers.     

The Left Hilbert BVP for h-Regular Functions in Clifford Analysis

Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e ₁, e ₂, . . . , e _n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ .     

On a problem in probability theory

For continuous random variables, we study a problem similar to that considered earlier by one of the authors for discrete random variables. Let numbers $$N > 0, E > 0, 0 \leqslant \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _s $$ be given. Consider a random vector x = (x ₁, …, x _s), uniformly distributed on the set $$x_j \geqslant 0, j = 1, \ldots ,s; \sum\limits_{j = 1}^s {x_j = N} , \sum\limits_{j = 1}^s {\lambda _j x_j \leqslant E} .$$ We study the weak limit of x as s → ∞.     

On the error term for the counting functions of finite Abelian groups

Leta(n) denote the number of non-isomorphic Abelian groups withn elements, and Δ(x) (resp. Δ _x ) appropriate error terms in the asymptotic formulas for the counting function \(\sum\nolimits_{n \leqslant x} {a(n)} (resp. \sum\nolimits_{m n \leqslant x} {a(m)} a(n))\) . Sharp bounds for $$\int\limits_1^X {\Delta (x) dx} , \int\limits_1^X {\Delta _{ 1} (x) dx} ,\int\limits_1^X {\Delta _1^2 (x) dx} $$ are given by using results on power moments of the Riemann zeta-function.     

Solvability of nonlinear systems including (γ, δ)-comparison pairs

Let $\gamma ,\delta \in \mathbb{R}^n $ with $\gamma _j ,\delta _j \in \{ 0,1\} $ . A comparison pair for a system of equations f_i(u₁,…,u_n)=0 (i=1,…,n) is a pair of vectors $v,w \in \mathbb{R}^n ,v \leqslant w$ , such that $$\begin{array}{*{20}c} {\gamma _i f_i (u_1 , \ldots ,u_{i - 1} ,v_i ,u_i + 1, \ldots ,u_n ) \leqslant 0,} \\ {\delta _i f_i (u_1 , \ldots ,u_{i - 1} ,w_i ,u_i + 1, \ldots ,u_n ) \geqslant 0} \\ \end{array} $$ for $\gamma _j u_j \geqslant v_j ,\delta _j u_j \leqslant w_j (j = 1, \ldots ,n)$ . The presence of comparison pairs enables one to essentially weaken the assumptions of the existence theorem. Bibliography: 1 title.     

Best quadrature formula on the class W * r L2 of periodic functions

For the classes of periodic functions with r-th derivative integrable in the mean,we obtain a best quadrature formula of the form $$\begin{gathered} \int_0^1 {f(x)dx = \sum\nolimits_{k = 0}^{m - 1} {\sum\nolimits_{l = 0}^\rho {p_{k,l} } } } f^{(l)} (x_k ) + R(f),0 \leqslant \rho \leqslant r - 1, \hfill \\ 0 \leqslant x_0< x_1< ...< x_{m - 1} \leqslant 1, \hfill \\ \end{gathered}$$ where ρ=r?2 and r?3, r=3, 5, 7, ..., and we determine an exact bound for the error of this formula.     

Asymptotic behavior of multiperiodic functionsG(x) = \prod\limits_{n = 1}^\infty {g(x/2^n )}

Let 0≤g be a dyadic Hölder continuous function with period 1 and g(0)=1, and let $G(x) = \prod\nolimits_{n = 0}^\infty {g(x/{\text{2}}^n )} $ . In this article we investigate the asymptotic behavior of $\smallint _0^{\rm T} \left| {G(x)} \right|^q dx$ and $\frac{1}{n}\sum\nolimits_{k = 0}^n {\log g(2^k x)} $ using the dynamical system techniques: the pressure function and the variational principle. An algorithm to calculate the pressure is presented. The results are applied to study the regulatiry of wavelets and Bernoulli convolutions.     

Degenerate differential operators in weighted Hölder spaces

A differential operator ?, arising from the differential expression $$lv(t) \equiv ( - 1)^r v^{[n]} (t) + \sum\nolimits_{k = 0}^{n - 1} {p_k } (t)v^{[k]} (t) + Av(t),0 \leqslant t \leqslant 1,$$ , and system of boundary value conditions $$P_v [v] = \sum\nolimits_{k = 0}^{n_v } {\alpha _{vk} } r^{[k]} (1) = 0.v - 1, \ldots ,\mu ,0 \leqslant \mu< n$$ is considered in a Banach space E. Herev ^[k](t)=(a(t) d/dt) ^k v(t)a(t) being continuous fort?0, α(t) >0 for t > 0 and \(\int_0^1 {\frac{{dz}}{{a(z)}} = + \infty ;}\) the operator A is strongly positive in E. The estimates , are obtained for ?: n even, λ varying over a half plane.     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

Mean value of weyl sums

We give a simple proof of a mean value theorem of I. M. Vinogradov in the following form. Suppose P, n, k, τ are integers, P≥1, n≥2, k≥n (τ+1), τ≥0. Put $$J_{k,n} (P) = \int_0^1 \cdots \int_0^1 {\left| {\sum\nolimits_{x = 1}^P {e^{2\pi i(a_1 x + \cdots + a_n x^n )} } } \right|^{2k} da_1 \ldots da_n .} $$ Then $$J_{k,n} \leqslant n!k^{2n\tau } n^{\sigma n^2 u} \cdot 2^{2n^2 \tau } P^{2k - \Delta } ,$$ where $$\begin{gathered} u = u_\tau = min(n + 1,\tau ), \hfill \\ \Delta = \Delta _\tau = n(n + 1)/2 - (1 - 1/n)^{\tau + 1} n^2 /2. \hfill \\ \end{gathered} $$      

On a Composition Preserving Inequalities between Polynomials

The Schur-Szegö composition of two polynomials \(f\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{z^j}} \) and \(g\left( z \right) = \sum\nolimits_{j = 0}^n {{B_j}{z^j}} \), both of degree n, is defined by \(f * g\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{B_j}{{\left( {\begin{array}{*{20}{c}}n \\ j \end{array}} \right)}^{ - 1}}{z^j}} \). In this paper, we estimate the minimum and the maximum of the modulus of f * g(z) on z = 1 and thereby obtain results analogues to Bernstein type inequalities for polynomials.